Having proved the main results, we consider a number of applications of the DGSPT formalism developed in Sections -. We begin in Section REF  with a geometric analysis of a two-dimensional map-based model for neuronal dynamics originally presented in [1]}. This example will help to demonstrate the application of the theory, and is intended to provide a kind of `benchmark application' similar to the Van der Pol oscillator [2]}, [3]} for continuous-time fast-slow ODE systems. Sections REF -REF  are more theoretical, and demonstrate the utility of the theory for analysing discretizations and Poincaré maps induced by fast-slow ODE systems respectively.
